Factorials

Hello. Here is a short guide on factorials.


The Factorial

Before going into the definition of factorials. Here are some examples of factorials.

    \[3! = 3 \times 2 \times 1 = 1 \times 2 \times 3 = 6\]

    \[5! = 5 \times 4 \times 3 \times 2 \times 1 = 1 \times 2 \times 3 \times 4 \times 5 = 120\]

As you can see, a factorial is a compact way of expressing the multiplication of descending (or ascending) positive whole numbers.

With factorials, the general case is:

    \[n! = n \times (n - 1) \times (n - 2) \times ... \times 1\]

or

    \[n! = 1 \times 2 \times ... \times (n -1) \times n\]

for n = {2, 3, 4, 5, ...}.

There are also some special cases with 1! = 1 and 0! = 1.


Division With Factorials

You may encounter fractions with factorials in the numerator and in the denominator. Here are some examples.

Example One

    \[\dfrac{4!}{2!} = \dfrac{4 \times 3 \times 2 \times 1}{2 \times 1} = 4 \times 3 = 12\]

Example Two

    \[\dfrac{10!}{9!} = \dfrac{10 \times 9!}{9!} = 10\]

Example Three

    \[\dfrac{8!}{4! \times 2!} = \dfrac{8 \times 7 \times 6 \times 5 \times 4!}{4! \times 2!} = \dfrac{8 \times 7 \times 6 \times 5 }{2} = 7 \times 6 \times 5 \times 4 = 840\]

Example Four

    \[\dfrac{4! \times 5!}{6!} = \dfrac{4! \times 5 \times 4 \times 3 \times 2 \times 1}{6 \times 5 \times 4!} = \dfrac{4 \times 3 \times 2 \times 1}{6} = \dfrac{24}{6} = 4\]

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